Coulomb-quantum oscillator correspondence in two dimension, pure gauge field and half-quantized vortex

Isotropic oscillator and Coulomb problems are known to have interesting correspondence. We focus on two-dimensional (2D) quantum problems and present complete treatment on the correspondence including the Schrödinger equation, eigenfunctions and eigenvalues, and the integrals of motion. We find only partial equivalence. The wave function correspondence is examined introducing local gauge transformation and the emergence of half-quantized vortex with the associated spin-half is established. Vortex structure of the electron proposed by us and the origin of charge are discussed in this paper. Outlook on the implications for QCD and hadron spectrum is outlined.